Three mutually perpendicular lines whose direction ratios are l1,m1,n1;l2,m2,n2;l3,m3,n3then prove that a given straight line whose direction ratio l1+l2+l3;m1+m2+m3;n1+n2+n3;make equal angles of these lines

Because the first three lines are mutually perpendicular, dot products of pairs of lines are zero. 
l1*l2+m1*m2+n1*n2 = 0 
l2*l3+m2*m3+n2*n3 = 0 
l1*l3+m1*m3+n1*n3 = 0 

Finally, using the above equations, and another dot product, it is shown that the cosine of the angle between the fourth vector and any of the first three vectors is the same.

Hence, it makes equal angles with them